All you need is flat bars and trigger shifters that are compatible with your derailleur's cable pull and 10spd cog set. Not very expensive and relatively simple.

Buying a new bike is not more cost effective than spending $35 on bars, $100 on shifters and whatever the shop charges you to install. Plus, if you're willing to sell your Ultegra brifters, you could easily get ~$120 bucks back from selling on ebay or CL if they're in decent shape.